事件空间中二阶非Четаев型非完整系统的守恒律

研究事件空间中二阶非чeTaeB型非完整系统的守恒律。提出事件空间中的Jourdain原理，引入事件空间中的Jorudain生成元，给出无限小变换下的Jourdain原理的不变性条件。在一定条件下得到事件空间中系统的守恒律。并举例说明结果的应用。     

一维双曲守恒方程组的Taylor-Galerkin有限元方法

１．引言 在文章［８］中,利用双曲守恒律的Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程形式,应用 Ｇａｌｅｒｋｉｎ有限元给出了求解一维双曲守恒律的计算方法．不同于间断有限元方法［２］、［３］和 Ｔａｙｌｏｒ－Ｇａｌｅｒｋｉｎ有限元方法［１］求解双曲守恒律,文章［８］采用连续 Ｇａｌｅｒｋｉｎ有限元求解双曲守恒律． 在文章［８］中,对差分方法和有限元方法求解双曲守恒律作了较为详细的讨论．同时在文章［８］中,采用积分变换,将双曲守恒律方程变成 Ｈａｍｉｌｔｏｎ－Ｊａｃｏｂｉ方程形式．由于 Ｈａｍｉｌｔｏｎ－Ｊａｃｏ…     

非凸单个守恒律初边值问题的整体弱熵解的构造

本文研究具有两段常数的初始值和常数边界值的非凸单个守恒律的初边值问题.在流函数具有一个拐点的条件下,由相应的初始值问题弱熵解的结构和Bardos-Leroux-Nedelec提出的边界熵条件,给出初边值问题整体弱熵解的一个构造方法,澄清弱熵解在边界附近的结构.与严格凸的单个守恒律初边值问题相比,非凸单个守恒律初边值问题的弱熵解中包括下列新的相互作用类型:一个接触或非接触激波碰到边界,边界弹回一个非接触激波.     

广义力学中完整非保守系统的Noether守律

本给出广义学中完整非保守系统三种形式的Ｎｏｅｔｈｅｒ守恒律。     

非完整力学系统“几何化”处理的新途径与可解性研究

本文提出一种新的几何化处理非完整力学系统的方法,并首次在一般几何框架下探讨了可解性问题,给出了一系列可解性条件。文中给出的框架本质上不同于Hermann等人给出的途径。文中还全面讨论了Hermann等人框架的缺陷,并证明了在Hermann框架下仅当约束由相应的无约束系统的自然守恒律给出时约束系统才是可解的。     

零压相对论欧拉方程组含有狄拉克函数初值的黎曼问题

研究一维零压相对论欧拉双曲守恒律系统含有狄拉克函数的初值条件的黎曼问题.借助特征线分析方法,求出了四种不同情形下的整体广义解,包括了含有狄拉克激波.     

一类演化方程的三个基本守恒律

“Soliton”的发现是当代数学物理的一大进展,它和与此有关的散射反演方法之产生,对非线性方程求解起了很大的推动作用。具Soliton解之系统的特点之一是存在无穷多个守恒律,多年来人们认为无穷多个守恒律的存在乃是孤子解存在的必要条件,无论对经典力学还是量子力学,守恒律的存在反映了系统对称性,且对称性比守恒律更为普遍。所     

具有非凸标量守恒律的解的衰减性

我们研究了具有周期初值数据的标量守恒律的方程的解的大时间行为,在很弱的非线性条件下,我们证明了当时间趋于无穷时其解收敛于一常数,本文的结果改进了以前的结果,因为我们只需在初值数据的平均值处流量是非线性的。     

变质量高阶非Четаев型非完整约束系统动力学

本文给出了高阶非型约束加在广义虚位移上的限制条件，建立了变质量高阶非型非线性非完整系统的Ｒｏｕｔｈ方程、方程、Ｎｉｅｌｓｅｎ方程和Ａｐｐｅｌｌ方程；给出了高阶非型约束系统“ｄ”与“δ”之间的交换关系，建立了其积分变分原理；并得到了变质量高阶非型约束系统的广义Ｎｏｅｔｈｅｒ守恒律．     

一类非严格双曲型守恒律整体连续解的存在性

该文利用初值扰动法，讨论了一类非严格双曲型守恒律Ｃａｕｃｈｙ问题整体连续解的存在性．这类方程包括了等熵气体动力学方程组．一个典型的变形方程组，弹性力学方程组，二次流守恒律和旋转退化方程等有意义的模型．     

Dynamics of quantum entropy maximization for finite-level systems

The problem of constructing models for the statistical dynamics of finite-level quantum mechanical systems is considered. The maximum entropy principle formulated by E.T. Jaynes in 1957 and asserting that the entropy of any physical system increases until it attains its maximum value under constraints imposed by other physical laws is applied. In accordance with this principle, the von Neumann entropy is taken for the objective function; a dynamical equation describing the evolution of the density operator in finite-level systems is derived by using the speed gradient principle. In this case, physical constraints are the mass conservation law and the energy conservation law. The stability of the equilibrium points of the system thus obtained is investigated. By using LaSalle’s theorem, it is shown that the density function tends to a Gibbs distribution, under which the entropy attains its maximum. The method is exemplified by analyzing a finite system of identical particles distributed between cells. Results of numerical simulation are presented.     

New conservation laws obtained directly from symmetry action on a known conservation law

Two formulas are introduced to directly obtain new conservation laws for any system of partial differential equations from a known conservation law and admitted symmetries. The first formula maps any conservation law of a given system to the corresponding conservation law of the system obtained through a contact transformation. When the contact transformation is a symmetry of the given system, then the corresponding conservation law is a conservation law of the given system. The second formula checks a priori whether or not the action of a symmetry (continuous or discrete) on a conservation law can yield one or more new conservation laws of the given system. Several examples are considered, including the use of a discrete symmetry to obtain a new conservation law and the use of a continuous symmetry to generate two new conservation laws.     

由虚功原理推导J积分守恒

本文由力学的普遍原理——虚功原理推导出了三维情况下J积分守恒的一般形式;十分明显地给出了J积分守恒成立的几个必要条件.     

扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用

首先,我们给出了引入伴随方程(组)扩充原方程(组)的策略使给定偏微分方程(组)的扩充方程组具有对应泛瓯即,成为Lagrange系统的方法,以此为基础提出了作为偏微分方程(组)传统守恒律和对称概念的一种推广-偏微分方程(组)扩充守恒律和扩充对称的概念;其次,以得到的Lagrange系统为基础给定了确定原方程(组)扩充守恒律和扩充对称的方法,从而达到扩充给定偏微分方程(组)的首恒律和对称的目的;第三,提出了适用于一般形式微分方程(组)的计算固有守恒律的方法;第四,实现以上算法过程中,我们先把计算(扩充)守恒律和对称问题均归结为求解超定线性齐次偏微分方程组(确定方程组)的问题.然后,对此关键问题我们提出了用微分形式吴方法处理的有效算法;最后,作为方法的应用我们计算确定了非线性电报方程组在内的五个发展方程(组)的新守恒律和对称,同时也说明了方法的有效性.     

带松弛项的单个守恒律方程解的时态渐近性质

本文研究一维空间中带松弛项的单个守恒律方程解的大时间状态估计.在松弛项满足耗散条件下通过对线性化方程Green函数的逐点估计得到方程解在时间充分大时的衰减估计,并由此反映出“弱”惠更斯原理.     

A continuous/discrete fractional Noether’s theorem

We prove a fractional Noether’s theorem for fractional Lagrangian systems invariant under a symmetry group both in the continuous and discrete cases. This provides an explicit conservation law (first integral) given by a closed formula which can be algorithmically implemented. In the discrete case, the conservation law is moreover computable in a finite number of steps.     

Numerical Solution of a Scalar One-Dimensional Monotonicity-Preserving Nonlocal Nonlinear Conservation Law

In this paper, we present numerical studies of a recently proposed scalar nonlocal nonlinear conservation law in one space dimension. The nonlocal model accounts for nonlocal interactions over a finite horizon and enjoys maximum principle, monotonicity-preserving and entropy condition on the continuum level. Moreover, it has a well-defined local limit given by a conventional local conservation laws in the form of partial differential equations. We discuss convergent numerical approximations that preserve similar properties on the discrete level. We also present numerical experiments to study various limiting behavior of the numerical solutions.     

Local limit of nonlocal traffic models: Convergence results and total variation blow-up

Consider a nonlocal conservation law where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit obtained by letting the convolution kernel converge to the Dirac delta one formally recovers a conservation law. However, recent counter-examples show that in general the solutions of the nonlocal equations do not converge to a solution of the conservation law. In this work we focus on nonlocal conservation laws modeling vehicular traffic: in this case, the convolution kernel is anisotropic. We show that, under fairly general assumptions on the (anisotropic) convolution kernel, the nonlocal-to-local limit can be rigorously justified provided the initial datum satisfies a one-sided Lipschitz condition and is bounded away from 0. We also exhibit a counter-example showing that, if the initial datum attains the value 0, then there are severe obstructions to a convergence proof.     

非完整非有势系统相对于非惯性系的广义Noether定理

本文构造了力学系统相对运动的Lagrange函数,建立了非线性非完整非有势系统相对于非惯性系的Jourdain型变分原理,提出并证明了这类力学系统相对于非惯性系的广义Noether定理,研究了其守恒量.